L(s) = 1 | − i·7-s + 3·11-s − 4i·13-s − 6i·17-s + 4·19-s − 3i·23-s + 3·29-s − 10·31-s + 7i·37-s − i·43-s − 12i·47-s − 49-s − 6i·53-s + 12·59-s − 4·61-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 0.904·11-s − 1.10i·13-s − 1.45i·17-s + 0.917·19-s − 0.625i·23-s + 0.557·29-s − 1.79·31-s + 1.15i·37-s − 0.152i·43-s − 1.75i·47-s − 0.142·49-s − 0.824i·53-s + 1.56·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.637929853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637929853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68519459455951485668326698699, −7.11560338521717918413242425983, −6.56603745670835960160324422235, −5.49030598156504651563292690032, −5.10196232464633745481516783849, −4.08762263087621523526595477272, −3.34950228022828918976509199505, −2.60224800943285292542151671419, −1.35017779617208586007762528395, −0.42757675158975702221547456367,
1.33645845069426978825235813149, 1.95178073243425631827428459389, 3.15503970886952016409268439764, 3.93222993704373413354317890318, 4.51058905661888656209865764273, 5.64973004598675539874019358222, 6.01781934009883028351677982200, 6.92111606218338561244311284497, 7.46427488600430883842200283114, 8.322096589156685691598145824695